Topological classification of nodal-line semimetals in square-net materials

Fermi surface topology and Berry phase of square-net-layered materials with P4/nmm space group were classified based on the first-principles analysis of Berry phases in each materials. The variation of electron-hole (e-h) asymmetry, driven by the third-nearest-neighbor hoppings of pnictogen ${p}_{x,y}$ orbitals in square-net, is essential to yield various topological Fermi surfaces. The change of chemical potential too yields various topological Fermi surfaces. By determining e-h asymmetry and chemical potential of the reported square-net-layered materials, we construct the phase diagram of the Fermi surface topologies. In addition, we adopt interlayer interaction and investigate three-dimensional Fermi surface. We classify the three-dimensional Fermi surface topologies based on the phase diagram, and our classification reproduces the quantum oscillation and spectroscopy results for various square-net compounds.